A mean variance approach to fisheries ma
Proceedings of the International Conference on Applied Computer Science
A mean-variance approach to
fisheries management
Marius Radulescu, Constanta Zoie Radulescu, Magdalena Turek Rahoveanu
an integrated approach to fisheries within ecologically
meaningful boundaries.
The ecosystem approach to fishery management is a
significant step towards sustainable use of the environment.
An approach to fish management that is compatible with the
ecosystem-based approach is the portfolio theory.
A portfolio framework systematically combines fish stocks
that are joined by ecology (e.g., predation, competition) and
unspecialized fishing technologies (e.g., mixed-species trawls)
into a portfolio which balances expected aggregate returns
against the risks associated with stock-attribute and other
uncertainties. To be productive, however, this framework
must be combined with property rights institutions that clearly
state management objectives, create long-run time-horizons
among harvesters, internalize spillovers caused by ecological
and technological jointness, and reduce uncertainty through
research and adaptive management. Although the cost of
reducing scientific uncertainty about ecological interactions
may limit the portfolio approach to intensive management of
relatively few species, its scope can be broadened to integrate
tradeoffs among more types of marine resources, such as
nature preserves and oil and gas deposits.
In the present paper we present several mean-variance
portfolio selection models for fish management: one multiobjective model with two objective functions and three single
objective models (a minimum risk model, a maximum
expected return model and an optimal tradeoff model). The
minimum risk model is similar to that presented in [1] with the
exception of an additional constraint connected to the budget
for fish harvesting. Also in our model we determine the range
of variation for the parameter W and for the budget of the
harvesting plan. A numerical example for a fish farm of semiintensive type, located in Jurilovca village, Tulcea county,
Romania is discussed. An efficient frontier of the minimum
risk problem is computed and displayed.
Abstract— A traditional approach for the fish management is the
single-species approach. In the last decades a new approach has
gained increased recognition. This is the ecosystem-based approach.
An approach to fish management that is compatible with the
ecosystem-based approach is the portfolio theory. In the present
paper we present several mean-variance portfolio selection models
for fish management. One of the models is the minimum risk model.
For this model we determine the range of variation for the parameter
W (that represents a lower limit for the expected return) and for the
budget of the harvesting plan. A numerical example for a fish farm of
semi-intensive type, located in a village from the Danube Delta,
Romania is discussed. An efficient frontier of the minimum risk
problem is computed and displayed.
Keywords: fisheries management, portfolio theory, meanvariance model, multi-objective programming problem.
I. INTRODUCTION
0
A
traditional approach for the fish management is the
single-species approach. Since the 1990s, fisheries
managers have been advised to broaden their scope of
awareness beyond single-species considerations owing to:
- general poor performance of single-species fishery
management worldwide.
- heightened awareness of interactions among fisheries and
ecosystems.
- better understanding of the functional value of ecosystems to
humans.
- recognition of the wide range of societal objectives
associated with fishery resources and ecosystems.
As a result, fisheries management has been shifting toward
an ecosystem-based fisheries management, also called an
ecosystem approach to fisheries (EAF). EAF strives to
balance diverse societal objectives by taking into account the
knowledge and uncertainties of biotic, abiotic, and human
components of ecosystems and their interactions and applying
II. MATHEMATICAL MODELS APPLIED IN FISHERIES
1
MANAGEMENT
Fisheries management is one of the fields where
mathematical models of operations research were first used
and also where they have been most widely applied. The
number of mathematical models in this domain has rapidly
grown in the last decades, due to the impressive development
of personal computers and commercial software programs.
Mathematical programming, differential equations, optimal
control, decision theory, neural networks, probability theory,
Manuscript received 15 july 2010. This work was supported by the
National Center for Program Management under PN II Proiect 1622 (2008 2011), Contract: 52123/2008, Management Information System for farms
fisheries in the South East Region with implications on the market.
M. Radulescu is with the Institute of Mathematical Statistics and Applied
Mathematics, Bucharest,ROMANIA(e-mail:mradulescu.csmro@yahoo.com).
C. Z. Radulescu is with the National Institute for R&D in Informatics,
Bucharest, ROMANIA (e-mail: radulescu@ ici.ro).
M. Turek Rahoveanu, is with the Research Institute for Agricultural
Economics and Rural Development, Bucharest, ROMANIA (e-mail:
mturek2003@yahoo.com).
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Proceedings of the International Conference on Applied Computer Science
statistics represent a modest enumeration of disciplines that
contribute to the improvement of fisheries management.
Applications of mathematical models in the fisheries
management include analysis of optimal management
decisions with regard to target stock levels over time, optimal
employment of fishing effort and capital, and cost minimizing
rules for monitoring and surveying stocks, or for enforcing
fishing regulations.
The great majority of the problems connected to fisheries
management have a multi-criteria character.
Well-defined objectives are obviously a prerequisite for
sensible management of fisheries. Many objectives for
fisheries have been suggested (see e.g. [2], and references
therein). Among the most frequently mentioned objectives
are: (i) maximum employment, (ii) maintaining regional
habitation, (iii) maximum sustainable yield (MSY), (iv)
conservation of fish stocks and the environment, (v)
generation of exports and foreign exchange (vi) economic
efficiency, i.e. maximum economic rents and (vii) social
equity. Clearly, not all of these objectives are independent.
Multi-objective decision analysis (MDA) is a useful
assessment method when fishery managers need a systematic
investigation of the tradeoffs involved in the selection of
alternative policy options. An important class of techniques
within MDA is vector optimization, consisting of
mathematical programming models with vector valued
objective functions. From the management perspective, vector
optimization models are particularly suited for situations when
the decision rule requires each objective to be kept as high (or
low) as possible. Solving vector optimization problems
usually entails finding a set of Pareto-optimal solutions. These
solutions are relevant to the decision-making process if
decision-makers have monotonic preferences. In [3] a vector
optimization model of the Eastern Pacific yellowfin tuna
fishery is used to generate Pareto-optimal solutions and to
evaluate the tradeoffs (shadow prices) involved in the
selection of alternative policy options.
Three conflicting biological objectives are considered: (a)
minimizing dolphin incidental mortality, (b) minimizing bycatch of all non-dolphin species and (c) maximizing total
yellowfin tuna catch. Results are presented and discussed by
means of non-linear tradeoff curves.
The traditional method to deal with a multi-objective
optimization problem is to define one or several single
objective optimization problems starting from the initial
problem and to solve them. The fisheries management implies
choice. For instance the basic choices are (i) which
management system to adopt and (ii) what management
measures to select. Logic dictates that it is either possible to
make a choice or it is not. In the latter case, management will
not be possible so there is no reason to waste valuable
resources on research. The first case, where choice is possible,
is more interesting.
In [4] the sustainable management of natural resources, and
in particular of fisheries is studied. In their research the
authors must take into account several conflicting objectives.
This is the case in the French Guiana shrimp fishery for which
profitability objectives imply a reduction in the fishing
activity. On the one hand, this fishery has negative
externalities on marine biodiversity due to discards. On the
other hand, this fishery has positive externalities on the
economy of the local community and interestingly enough on
a protected seabird species in the area (the Frigatebird that
feeds on discards). In their paper the authors have several
sustainability objectives: an economic objective in terms of
the profitability of the fishing activity, and a conservation
objective in terms of the Frigatebird population. For that
purpose, they have developed a dynamic model of that
bioeconomic system and study the trade-offs between the two
conflicting objectives. Their study provides a means for the
quantification of the economic and ecological objectives that
ensure a viable management solution. In the same time their
study leads to the development of new tools for the arbitration
of conflicting sustainability objectives. In particular, such
tools could be used as a quantitative basis for cost–benefit
analysis taking into account environmental externalities.
In [5] the potential of choice modeling for evaluating the
importance of potentially conflicting fisheries management
objectives is examined. The fisheries of the English Channel
are used as an example. Results from a survey of the key
stakeholders in these fisheries are presented, showing that
regional employment and sustainable yields are of key
concern. Overall, the importance between objectives can be
measured appropriately and the methodology can offer useful
information to the management process.
In [6] is proposed a model that considers a long-term
planning horizon and specifically allows an optimal total
allowable catch quota to be obtained for the first planning
period. A formulation and algorithmic resolution of a twostage stochastic nonlinear programming model with recourse
is presented. This model takes into account biomass dynamics,
the conditions guaranteeing sustainable species management
and uncertain parameters such as growth rate and species
carrying capacity. These parameters are explicitly
incorporated via a discrete random variable (scenarios). The
proposed model is solved by Lagrangian decomposition using
the algebraic modeling software AMPL, in combination with
the solver MINOS to solve the nonlinear models resulting
from the scenario decomposition.
In [7] is presented a modeling framework for assessing the
implication of long term fisheries policy decisions. It
illustrates an approach to model building that begins with
establishment of a matrix reflecting the basic features of the
fisheries sector and then proceeds to incorporate these features
into an integer programming model. The model estimates the
real cost of alternative policies and takes into account the
various biological and economic constraints to production.
The model has provided a valuable tool for the assessment of
different policy options for Kuwait's fisheries sector. The
options involved establish the most appropriate way to
maximize fresh-fish self-sufficiency.
Dynamic programming (DP) has shown itself to be an
appropriate methodology for devising strategies of fisheries
management. In [8] are suggested several improvements for
the dynamic programming models such as backward
formulations for more realistic decisions in both the cases of
invertible and non-invertible transformations; the introduction
of varying catch capacity and associated costs for changes and
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Proceedings of the International Conference on Applied Computer Science
for underutilized capacity; the relevance of maxi-min
objectives; and the computational implications of multispecies
models.
In [9] are presented applications of neural networks in
fisheries management. Some of these applications are
connected to forecasting in fisheries that covers the
distribution of eggs, recruitment, fish growth/age, biomass and
fish catch. Other major areas are identification, abundance and
food products, environmental factors and collapse of fishery
industry. The data structures are given in tensorial notation.
In [10] the goal programming was used to optimize the
achievement of Sri Lanka's fisheries development targets for
the planning period 1988–1991.
In [11] were used mathematical programming methods in
conjunction with multilevel planning concepts, to estimate the
value of foreign access to U.S. fisheries.
An important mathematical instrument which was
successfully applied to modeling the problems from fisheries
management is portfolio theory. The above mentioned theory
was developed as a result of the research in the domain of
financial management. Its aim is the elaboration of a
quantitative analysis of how investors can diversify their
portfolio in order to minimize risk and maximize returns. The
theory was introduced in 1952 by University of Chicago
economics student Harry Markowitz, who published his
doctoral thesis, “Portfolio Selection” in the Journal of Finance
[12]. There exist many applications of portfolio theory to
domains that do not imply finance such as agriculture, energy,
biodiversity etc. For references regarding applications of
portfolio theory to non-financial areas see [13] and [14]. The
application of portfolio theory for finding an optimal
harvesting portfolio is popular in the literature.
Managing fish stocks in terms of a portfolio of economic
assets is likely to significantly increase benefits for society
relative to single-species approaches. A portfolio framework
systematically combines fish stocks that are joined by ecology
(e.g., predation, competition) and unspecialized fishing
technologies (e.g., mixed-species trawls) into a portfolio
which balances expected aggregate returns against the risks
associated with stock-attribute and other uncertainties. In [15]
the authors note that in order to be productive, this framework
must be combined with property rights institutions that clearly
state management objectives, create long-run time-horizons
among harvesters, internalize spillovers caused by ecological
and technological jointness, and reduce uncertainty through
research and adaptive management. Although the cost of
reducing scientific uncertainty about ecological interactions
may limit the portfolio approach to intensive management of
relatively few species, its scope can be broadened to integrate
tradeoffs among more types of marine resources, such as
nature preserves and oil and gas deposits.
In [1] the authors adapt financial portfolio theory as a
method for ecosystem-based fishery management that
accounts for species interdependencies, uncertainty, and
sustainability constraints. They illustrate their method with
routinely collected data available from the Chesapeake Bay
and demonstrate the gains from taking into account variances
and covariances of gross fishing revenues in setting species
total allowable catches.
In [16] the authors embed a portfolio decision framework
into a multi-period bio-economic model in order to quantify
the risk-benefit tradeoffs of alternative strategies. They
develop alternative sets of processed seafood products for
managing the risks that occur as a result of harvests from
commercial fish stocks. The model is used to generate an
efficient portfolio frontier to estimate possible rent dissipation
from status quo management. Frontiers are also generated for
seafood processors and brokers. The authors discuss
implications for the different industry agents.
Drawing the analogy between managing risky assets and
managing multispecies fisheries is a relatively new idea, even
though the foundation for this idea is neither new to ecology
nor economics. In ecology, Walters at al. [17] derives a meanvariance frontier for single-species management, while Real
[18] uses portfolio theory to describe animal behavior.
Portfolio management of fisheries can be a means of
allocating catch across life history stages. Baldursson and
Magnusson [19], Arrason [20] alludes to multispecies
portfolio management in a deterministic bioeconomic model
by suggesting that managers choose a vector of Total
Allowable Catches (TACs), while Hanna [21] explicitly
discusses the idea of selecting “species portfolios” as a means
to match management objectives with ecosystem structure.
Hilborn et al. [22] provide a justification for portfolio
management at the regional level by pointing out that total
productivity aggregated across species is subject to less
variability than the productivity of individual species.
Edwards et al. [15] formally develop the analogy within the
context of standard bio-economic models and provide a
stylized simulation of a three-species system. More recently,
Perusso at al. [23] apply portfolio theory to individual
fishermen targeting decisions in the U.S. Atlantic and Gulf of
Mexico pelagic longline fishery. An interesting survey on the
application of portfolio theory to fisheries management may
be found in [24]. Several references on applications of
operations research to fish management can be found in [25],
[26].
III. MEAN-VARIANCE PORTFOLIO SELECTION MODELS FOR
2
FISHERIES MANAGEMENT
Suppose that we have n fish species S1 , S 2 ,..., S n . Denote
by ξ i the return obtained from one kg of fish of species S i .
Of course all ξ i are random variables. Let ξ = (ξ 1 , ξ 2 ,..., ξ n )
denote the random vector of returns. Denote by μ i the mean
of ξ i . Let μ = (μ1 , μ 2 ,..., μ n ) . Denote by bi the cost
associated to fishing one kg of fish from species S i and with
a i the maximum sustainable yield for the fish of species S i .
a = (a1 , a 2 ,..., a n ) , b = (b1 , b2 ,..., bn ) . Denote by
Let
C = cij the covariance matrix of the random vector
( )
(
)
ξ = (ξ 1 , ξ 2 ,..., ξ n ) . That is cij = cov ξ i , ξ j , 1 ≤ i, j ≤ n . The
signification of c ij
is the correlation between species
revenues. These correlations can be negative or positive
depending on the relative strength of trophic interactions,
environmental fluctuations, and fishing intensity and gear
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Proceedings of the International Conference on Applied Computer Science
choices that determine fish stocks and corresponding catch
rates, as well as output market interactions that affect prices.
Let x i be the quantity harvested from fish of species S i .
Of course x i ≥ 0 for every i=1,2,…,n. The vector
x = (x1 , x 2 ,..., x n ) will be called a portfolio. The cost of
n
harvesting a portfolio x is equal to
∑b x
i
i
B. The maximum expected return problem
In the frame of this problem the decision maker looks for a
portfolio x = (x1 , x 2 ,..., x n ) that maximizes the expected return
and satisfies the following constraints: the risk is smaller than
a given limit τ and the sum invested in the portfolio of fish is
equal to M.
⎧max μ T x
⎪ T
⎨x Cx ≤ τ
⎪ T
⎪⎩b x = M , 0 ≤ x ≤ a
An important problem is to find the range of the parameter
τ . In order to find the limits of this range we shall solve the
following problems
( τ min) min x T Cx : b T x = M ,0 ≤ x ≤ a
5
= b T x . The return
i =1
(resp. the expected return) from harvesting a portfolio x is
n
equal to
∑ξ x
i
i
= ξ T x (resp. to
i =1
n
∑μ x
i i
= μ T x ).
i =1
In reality the vectors a = (a1 , a 2 ,..., a n ) , ξ = (ξ 1 , ξ 2 ,..., ξ n )
and μ = (μ1 , μ 2 ,..., μ n ) and the matrix C = c ij depend on
( )
{
}
( τ max) max{ x Cx : b x = M ,0 ≤ x ≤ a}
time. Fishing and pollution are important factors that make
that the above mentioned vectors and matrix vary along the
time. For short periods of time one can consider that these
vectors and the covariance matrix are constant.
The financial risk of the portfolio x may be defined in
several ways. One traditional way is to define the risk as the
variance of the return f 1 (x ) = Var ξ T x = x T Cx .
A general problem in the fisheries management is the
following multi-objective problem
⎧minimize x T Cx
⎪
T
⎪maximize μ x
⎨
⎪subject to
⎪b T x = M , 0 ≤ x ≤ a
⎩
Starting from the above multi-objective problem one can
formulate the following single objective problems: the
minimum risk problem, the maximum expected return
problem, the optimal tradeoff problem.
T
Denote the optimal value of the problem ( τ min) with τ 1
and the optimal value of the problem ( τ max) with τ 2 . The
decision maker may choose the parameter τ in the range
[τ 1 ,τ 2 ] .
( )
C. The optimal tradeoff problem
6
For every α ∈ [0,1] consider the problem
( )
( )
{
D. Numerical example
In this section we shall study an application of the minimum
risk model for a fish farm of semi-intensive type, located in
the region Danube Delta, at village Jurilovca, Tulcea county,
Romania. The fishery area is 834 ha and 779 ha of it are
covered by water. We study 9 fish species: Carp, Sanger,
Novac, Pike, Catfish, Crucian, Perch, Bream and Roach. The
historical data on the fish market prices are from the period
2000-2008. The sum invested in the harvest plan is M=5000
RON. As a result of computation we find W min = 2.71 x 103
RON,
W max = 21.81 x 103 RON.
7
4
)
TABLE 1. THE MAXIMUM SUSTAINABLE YIELD (MSY) AND THE
HARVESTING COST FOR THE FISH SPECIES
Fish Type
MSY
Harvesting cost
Carp
One can easily see that 0 ≤ M ≤ b T a . An important problem
is to find the range of the parameter W. In order to find the
limits of this range we shall solve the following problems
(Wmin) min μ T x : b T x = M ,0 ≤ x ≤ a
{
max{μ
}
P(α ) min (1 − α )x T Cx − αμ T x : b T x = M ,0 ≤ x ≤ a
Every optimal portfolio of the above problem is called an
efficient portfolio.
A. The minimum risk problem
In the frame of this problem the decision maker looks for a
portfolio x = (x1 , x 2 ,..., x n ) that minimizes the financial risk
and satisfies the following constraints: the expected return is
greater than a given limit W and the sum invested in the
portfolio of fish is equal to M.
⎧min x T Cx
⎪ T
⎪μ x ≥ W
⎨ T
⎪b x = M
⎪0 ≤ x ≤ a
⎩
(
T
}
x = M ,0 ≤ x ≤ a}
T
x : bT
(Wmax)
Denote the optimal value of the problem (Wmin) with W1
and the optimal value of the problem (Wmax) with W2 . The
decision maker may choose the parameter W in the range
[W1 ,W2 ] .
294
in tons
in RON/tone
1
9120
Sanger
0.5
5920
Novac
0.7
5920
Pike
0.4
3650
Catfish
0.5
3650
Crucian
0.3
3650
Perch
0.3
3650
Bream
0.3
3650
Roach
0.3
3650
Proceedings of the International Conference on Applied Computer Science
TABLE 2. HISTORICAL DATA ON THE MARKET PRICES FOR THE FISH SPECIES (IN 103 RON /TONE)
Fish Type
2000
2001 2002
2003
2004
2005
2006
2007
2008
Carp
4.5
4.5
5.0
5.5
6.2
6.7
7.0
7.5
7.8
Sanger
2.5
2.5
2.8
2.8
3.0
3.2
3.5
3.8
5.0
Novac
2.5
2.5
2.8
2.8
3.0
3.2
3.5
3.5
5.0
Pike
4.5
5.0
5.5
6.5
6.5
7.0
7.0
7.0
7.5
Catfish
6.5
7.0
7.0
8.0
8.5
8.5
9.0
9.0
9.5
Crucian
1.8
2.0
2.2
2.2
2.4
2.6
2.8
3.0
3.5
Perch
9.5
9.5
9.5
10.0
10.0
11.0
11.0
11.0
13.0
Bream
1.8
1.8
2
2.2
2.2
2.5
2.7
2.8
3.2
Roach
1.8
1.8
2
2.2
2.2
2.5
2.7
2.8
3.2
TABLE 3. FISH PORTFOLIOS FOR VARIOUS VALUES OF THE PARAMETER W.
W
Nr.
1
Carp
2.71
0.16
Sanger
Novac
Pike
Catfish
0.00
0.59
0.00
0.00
Crucian
Perch
Bream
0.00
0.00
0.00
Roach
0.00
Fob
0.37
2
3.65
0.16
0.00
0.20
0.00
0.00
0.00
0.06
0.30
0.30
0.42
3
4.60
0.17
0.00
0.11
0.00
0.00
0.00
0.17
0.30
0.30
0.52
4
5.54
0.19
0.00
0.02
0.00
0.00
0.00
0.28
0.30
0.30
0.62
5
6.49
0.13
0.00
0.00
0.00
0.14
0.00
0.30
0.30
0.30
0.75
6
7.44
0.03
0.00
0.05
0.00
0.31
0.00
0.30
0.30
0.30
0.90
7
8.38
0.00
0.00
0.00
0.00
0.47
0.00
0.30
0.30
0.30
1.06
8
9.33
0.00
0.00
0.00
0.20
0.50
0.00
0.30
0.30
0.07
1.30
9
10.27
0.00
0.00
0.00
0.40
0.50
0.00
0.30
0.24
0.00
1.62
10
11.22
0.01
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
2.10
11
12.17
0.17
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
2.66
12
13.11
0.33
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
3.28
13
14.06
0.48
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
3.98
14
15.00
0.64
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
4.74
15
15.95
0.79
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
5.58
16
16.89
0.95
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
6.48
17
17.84
1.00
0.00
0.00
0.40
0.50
0.25
0.30
0.30
0.30
7.45
18
18.79
1.00
0.00
0.26
0.40
0.50
0.30
0.30
0.30
0.30
8.53
19
19.73
1.00
0.00
0.55
0.40
0.50
0.30
0.30
0.30
0.30
9.70
20
20.68
1.00
0.15
0.70
0.40
0.50
0.30
0.30
0.30
0.30
11.00
21
21.62
1.00
0.44
0.70
0.40
0.50
0.30
0.30
0.30
0.30
12.40
14
12
Risk
10
8
6
4
2
0
2.71 3.65 4.6 5.54 6.49 7.44 8.38 9.33 10.3 11.2 12.2 13.1 14.1 15
16
16.9 17.8 18.8 19.7 20.7 21.6
W
Figure 1. The efficient frontier map associated to the minimum risk problem
In the second (resp. third) column from Table 1 is displayed
the vector a (resp. the vector b). In the Table 2 are presented
the historical data of the market prices for the fish species and
in the Table 3, the optimal fish portfolios for different values
295
Proceedings of the International Conference on Applied Computer Science
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[20] R. Arnason, “Ecological Fisheries Management Using Individual
Transferable Share Quotas”, Ecological Applications 8, 1998, pp: S151S159.
[21] S. Hanna, “Institutions for Marine Ecosystems: Economic Incentives and
Fishery Management”,Ecological Applications 8, 1998, pp: S170-S174.
[22] R. Hilborn, J-J. Maguire, A.M. Parma, A.A. Rosenberg, “The
Precautionary Approach and Risk Management: Can They Increase the
Probability of Successes in Fishery Management?” Canadian Journal of
Fisheries and Aquatic Sciences 58, 2001, pp: 99-107.
[23] L. Perruso, R. N. Weldon, and S. L. Larkin. "Predicting Optimal
Targeting Strategies in Multispecies Fisheries: A Portfolio Approach."
Marine Resource Economics 20, no. 1, 2005, pp: 25-45.
[24] M. M. Yang, “The portfolio approach for the ecosystem-based
fishery management”, Economics Department, University of Auckland,
PhD Research Proposal, April 2008.
www.nzares.org.nz/pdf/A%20Portfolio%20Approach.pdf
[25] R. Arnason, “Fisheries management and operations research”, European
Journal of Operational Research, Volume 193, Issue 3, 16 March 2009,
pp: 741-751.
[26] T. Bjørndal, D. E. Lane, A. Weintraub, “Operational research models
and the management of fisheries and aquaculture: A review”, European
Journal of Operational Research, Volume 156, Issue 3, 1 August 2004,
pp: 533-540.
of parameter W. In the last column of the Table 3 is displayed
the optimal value of the objective function (the variance of the
return).
In Figure 1 is displayed the graph of the efficient frontier
map associated to the minimum risk problem. That is to every
value of the parameter W corresponds the minimum value of
the objective map (the variance of the return).
The minimum risk problem was solved for various values
of the parameter W with the NLP solver from GAMS. The
limits of the range of variation of the parameter W were
obtained with the MINLP solver from GAMS.
3
IV.CONCLUSION
The paper studies an approach based on portfolio theory for
fisheries management. The numerical example presented is
based on real data. It shows that the portfolio theory is a
suitable approach for the fish management. A valuable idea
for further research is to study portfolio selection models
based on downside risk measures such as: semi-variance,
lower partial moments, probability shortfall or value-at-risk.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
J. N. Sanchirico, M. D. Smith, D. W. Lipton, “An empirical approach to
ecosystem-based fishery management”, Ecological Economics, Volume
64, Issue 3, 15, 2008, pp 586-596.
A.T. Charles, Sustainable Fishery Systems, Oxford U.K.: Blackwell
Science, 2001, 370 p.
R. R. Enríquez-Andrade, J. G. Vaca-Rodríguez, “Evaluating ecological
tradeoffs in fisheries management: a study case for the yellowfin tuna
fishery in the Eastern Pacific Ocean”, Ecological Economics, Volume
48, Issue 3, 31 March 2004, pp 303-315.
V. Martinet, F. Blanchard, “Fishery externalities and biodiversity:
Trade-offs between the viability of shrimp trawling and the conservation
of Frigatebirds in French Guiana”, Ecological Economics, Volume 68,
Issue 12, 15 October 2009, pp 2960-296.
P. Wattage, S. Mardle, S. Pascoe, “Evaluation of the importance of
fisheries management objectives using choice-experiments”, Ecological
Economics, Volume 55, Issue 1, 5 October 2005, pp 85-95.
V. M. Albornoz, C. M. Canales, “Total allowable catch for managing
squat lobster fishery using stochastic nonlinear programming”,
Computers and Operations Research, Volume 33, Issue 8 August 2006
pp: 2113 - 2124.
M Khorshid, “Fisheries development planning in Kuwait: An integer
programming model”, European Journal of Operational Research,
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L. E. Stanfel, “New dynamic programming models of fisheries
management”, Mathematical and Computer Modelling, Volume 10,
Issue 8, 1988, pp 593-607.
I. Suryanarayana, A. Braibanti, R. S. Rao, V. A. Ramam, D. Sudarsan,
G. N. Rao, “Neural networks in fisheries research”, Fisheries Research,
Volume 92, Issues 2-3, August 2008, pp 115-139.
P. Muthukude, J. L. Novak, C. Jolly, “A goal programming evaluation of
fisheries development plans for Sri Lanka's coastal fishing fleet, 1988–
1991”, Fisheries Research, Volume 12, Issue 4, December 1991, pp
325-339.
E. Meuriot and J. M. Gates, “Fishing Allocations and Optimal Fees: A
Single and Multilevel Programming”, Analysis, American Journal of
Agricultural Economics, Vol. 65, No. 4 Nov., 1983, pp. 711-721.
H.M. Markowitz, “Portfolio selection”, J.of Finance. 8, 1952, pp 77-91.
M. Radulescu, S. Radulescu, C.Z. Radulescu, Mathematical Models for
Optimal Asset Allocation. (Romanian). Editura Academiei Române,
Bucureşti, 2006.
M. Radulescu, C.Z. Radulescu, Gh. Zbaganu, “Asset allocation models
in discrete variable”, Studies in Informatics and Control, Volume 18,
Number 1, 2009, pp 63-70.
296
A mean-variance approach to
fisheries management
Marius Radulescu, Constanta Zoie Radulescu, Magdalena Turek Rahoveanu
an integrated approach to fisheries within ecologically
meaningful boundaries.
The ecosystem approach to fishery management is a
significant step towards sustainable use of the environment.
An approach to fish management that is compatible with the
ecosystem-based approach is the portfolio theory.
A portfolio framework systematically combines fish stocks
that are joined by ecology (e.g., predation, competition) and
unspecialized fishing technologies (e.g., mixed-species trawls)
into a portfolio which balances expected aggregate returns
against the risks associated with stock-attribute and other
uncertainties. To be productive, however, this framework
must be combined with property rights institutions that clearly
state management objectives, create long-run time-horizons
among harvesters, internalize spillovers caused by ecological
and technological jointness, and reduce uncertainty through
research and adaptive management. Although the cost of
reducing scientific uncertainty about ecological interactions
may limit the portfolio approach to intensive management of
relatively few species, its scope can be broadened to integrate
tradeoffs among more types of marine resources, such as
nature preserves and oil and gas deposits.
In the present paper we present several mean-variance
portfolio selection models for fish management: one multiobjective model with two objective functions and three single
objective models (a minimum risk model, a maximum
expected return model and an optimal tradeoff model). The
minimum risk model is similar to that presented in [1] with the
exception of an additional constraint connected to the budget
for fish harvesting. Also in our model we determine the range
of variation for the parameter W and for the budget of the
harvesting plan. A numerical example for a fish farm of semiintensive type, located in Jurilovca village, Tulcea county,
Romania is discussed. An efficient frontier of the minimum
risk problem is computed and displayed.
Abstract— A traditional approach for the fish management is the
single-species approach. In the last decades a new approach has
gained increased recognition. This is the ecosystem-based approach.
An approach to fish management that is compatible with the
ecosystem-based approach is the portfolio theory. In the present
paper we present several mean-variance portfolio selection models
for fish management. One of the models is the minimum risk model.
For this model we determine the range of variation for the parameter
W (that represents a lower limit for the expected return) and for the
budget of the harvesting plan. A numerical example for a fish farm of
semi-intensive type, located in a village from the Danube Delta,
Romania is discussed. An efficient frontier of the minimum risk
problem is computed and displayed.
Keywords: fisheries management, portfolio theory, meanvariance model, multi-objective programming problem.
I. INTRODUCTION
0
A
traditional approach for the fish management is the
single-species approach. Since the 1990s, fisheries
managers have been advised to broaden their scope of
awareness beyond single-species considerations owing to:
- general poor performance of single-species fishery
management worldwide.
- heightened awareness of interactions among fisheries and
ecosystems.
- better understanding of the functional value of ecosystems to
humans.
- recognition of the wide range of societal objectives
associated with fishery resources and ecosystems.
As a result, fisheries management has been shifting toward
an ecosystem-based fisheries management, also called an
ecosystem approach to fisheries (EAF). EAF strives to
balance diverse societal objectives by taking into account the
knowledge and uncertainties of biotic, abiotic, and human
components of ecosystems and their interactions and applying
II. MATHEMATICAL MODELS APPLIED IN FISHERIES
1
MANAGEMENT
Fisheries management is one of the fields where
mathematical models of operations research were first used
and also where they have been most widely applied. The
number of mathematical models in this domain has rapidly
grown in the last decades, due to the impressive development
of personal computers and commercial software programs.
Mathematical programming, differential equations, optimal
control, decision theory, neural networks, probability theory,
Manuscript received 15 july 2010. This work was supported by the
National Center for Program Management under PN II Proiect 1622 (2008 2011), Contract: 52123/2008, Management Information System for farms
fisheries in the South East Region with implications on the market.
M. Radulescu is with the Institute of Mathematical Statistics and Applied
Mathematics, Bucharest,ROMANIA(e-mail:mradulescu.csmro@yahoo.com).
C. Z. Radulescu is with the National Institute for R&D in Informatics,
Bucharest, ROMANIA (e-mail: radulescu@ ici.ro).
M. Turek Rahoveanu, is with the Research Institute for Agricultural
Economics and Rural Development, Bucharest, ROMANIA (e-mail:
mturek2003@yahoo.com).
291
Proceedings of the International Conference on Applied Computer Science
statistics represent a modest enumeration of disciplines that
contribute to the improvement of fisheries management.
Applications of mathematical models in the fisheries
management include analysis of optimal management
decisions with regard to target stock levels over time, optimal
employment of fishing effort and capital, and cost minimizing
rules for monitoring and surveying stocks, or for enforcing
fishing regulations.
The great majority of the problems connected to fisheries
management have a multi-criteria character.
Well-defined objectives are obviously a prerequisite for
sensible management of fisheries. Many objectives for
fisheries have been suggested (see e.g. [2], and references
therein). Among the most frequently mentioned objectives
are: (i) maximum employment, (ii) maintaining regional
habitation, (iii) maximum sustainable yield (MSY), (iv)
conservation of fish stocks and the environment, (v)
generation of exports and foreign exchange (vi) economic
efficiency, i.e. maximum economic rents and (vii) social
equity. Clearly, not all of these objectives are independent.
Multi-objective decision analysis (MDA) is a useful
assessment method when fishery managers need a systematic
investigation of the tradeoffs involved in the selection of
alternative policy options. An important class of techniques
within MDA is vector optimization, consisting of
mathematical programming models with vector valued
objective functions. From the management perspective, vector
optimization models are particularly suited for situations when
the decision rule requires each objective to be kept as high (or
low) as possible. Solving vector optimization problems
usually entails finding a set of Pareto-optimal solutions. These
solutions are relevant to the decision-making process if
decision-makers have monotonic preferences. In [3] a vector
optimization model of the Eastern Pacific yellowfin tuna
fishery is used to generate Pareto-optimal solutions and to
evaluate the tradeoffs (shadow prices) involved in the
selection of alternative policy options.
Three conflicting biological objectives are considered: (a)
minimizing dolphin incidental mortality, (b) minimizing bycatch of all non-dolphin species and (c) maximizing total
yellowfin tuna catch. Results are presented and discussed by
means of non-linear tradeoff curves.
The traditional method to deal with a multi-objective
optimization problem is to define one or several single
objective optimization problems starting from the initial
problem and to solve them. The fisheries management implies
choice. For instance the basic choices are (i) which
management system to adopt and (ii) what management
measures to select. Logic dictates that it is either possible to
make a choice or it is not. In the latter case, management will
not be possible so there is no reason to waste valuable
resources on research. The first case, where choice is possible,
is more interesting.
In [4] the sustainable management of natural resources, and
in particular of fisheries is studied. In their research the
authors must take into account several conflicting objectives.
This is the case in the French Guiana shrimp fishery for which
profitability objectives imply a reduction in the fishing
activity. On the one hand, this fishery has negative
externalities on marine biodiversity due to discards. On the
other hand, this fishery has positive externalities on the
economy of the local community and interestingly enough on
a protected seabird species in the area (the Frigatebird that
feeds on discards). In their paper the authors have several
sustainability objectives: an economic objective in terms of
the profitability of the fishing activity, and a conservation
objective in terms of the Frigatebird population. For that
purpose, they have developed a dynamic model of that
bioeconomic system and study the trade-offs between the two
conflicting objectives. Their study provides a means for the
quantification of the economic and ecological objectives that
ensure a viable management solution. In the same time their
study leads to the development of new tools for the arbitration
of conflicting sustainability objectives. In particular, such
tools could be used as a quantitative basis for cost–benefit
analysis taking into account environmental externalities.
In [5] the potential of choice modeling for evaluating the
importance of potentially conflicting fisheries management
objectives is examined. The fisheries of the English Channel
are used as an example. Results from a survey of the key
stakeholders in these fisheries are presented, showing that
regional employment and sustainable yields are of key
concern. Overall, the importance between objectives can be
measured appropriately and the methodology can offer useful
information to the management process.
In [6] is proposed a model that considers a long-term
planning horizon and specifically allows an optimal total
allowable catch quota to be obtained for the first planning
period. A formulation and algorithmic resolution of a twostage stochastic nonlinear programming model with recourse
is presented. This model takes into account biomass dynamics,
the conditions guaranteeing sustainable species management
and uncertain parameters such as growth rate and species
carrying capacity. These parameters are explicitly
incorporated via a discrete random variable (scenarios). The
proposed model is solved by Lagrangian decomposition using
the algebraic modeling software AMPL, in combination with
the solver MINOS to solve the nonlinear models resulting
from the scenario decomposition.
In [7] is presented a modeling framework for assessing the
implication of long term fisheries policy decisions. It
illustrates an approach to model building that begins with
establishment of a matrix reflecting the basic features of the
fisheries sector and then proceeds to incorporate these features
into an integer programming model. The model estimates the
real cost of alternative policies and takes into account the
various biological and economic constraints to production.
The model has provided a valuable tool for the assessment of
different policy options for Kuwait's fisheries sector. The
options involved establish the most appropriate way to
maximize fresh-fish self-sufficiency.
Dynamic programming (DP) has shown itself to be an
appropriate methodology for devising strategies of fisheries
management. In [8] are suggested several improvements for
the dynamic programming models such as backward
formulations for more realistic decisions in both the cases of
invertible and non-invertible transformations; the introduction
of varying catch capacity and associated costs for changes and
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Proceedings of the International Conference on Applied Computer Science
for underutilized capacity; the relevance of maxi-min
objectives; and the computational implications of multispecies
models.
In [9] are presented applications of neural networks in
fisheries management. Some of these applications are
connected to forecasting in fisheries that covers the
distribution of eggs, recruitment, fish growth/age, biomass and
fish catch. Other major areas are identification, abundance and
food products, environmental factors and collapse of fishery
industry. The data structures are given in tensorial notation.
In [10] the goal programming was used to optimize the
achievement of Sri Lanka's fisheries development targets for
the planning period 1988–1991.
In [11] were used mathematical programming methods in
conjunction with multilevel planning concepts, to estimate the
value of foreign access to U.S. fisheries.
An important mathematical instrument which was
successfully applied to modeling the problems from fisheries
management is portfolio theory. The above mentioned theory
was developed as a result of the research in the domain of
financial management. Its aim is the elaboration of a
quantitative analysis of how investors can diversify their
portfolio in order to minimize risk and maximize returns. The
theory was introduced in 1952 by University of Chicago
economics student Harry Markowitz, who published his
doctoral thesis, “Portfolio Selection” in the Journal of Finance
[12]. There exist many applications of portfolio theory to
domains that do not imply finance such as agriculture, energy,
biodiversity etc. For references regarding applications of
portfolio theory to non-financial areas see [13] and [14]. The
application of portfolio theory for finding an optimal
harvesting portfolio is popular in the literature.
Managing fish stocks in terms of a portfolio of economic
assets is likely to significantly increase benefits for society
relative to single-species approaches. A portfolio framework
systematically combines fish stocks that are joined by ecology
(e.g., predation, competition) and unspecialized fishing
technologies (e.g., mixed-species trawls) into a portfolio
which balances expected aggregate returns against the risks
associated with stock-attribute and other uncertainties. In [15]
the authors note that in order to be productive, this framework
must be combined with property rights institutions that clearly
state management objectives, create long-run time-horizons
among harvesters, internalize spillovers caused by ecological
and technological jointness, and reduce uncertainty through
research and adaptive management. Although the cost of
reducing scientific uncertainty about ecological interactions
may limit the portfolio approach to intensive management of
relatively few species, its scope can be broadened to integrate
tradeoffs among more types of marine resources, such as
nature preserves and oil and gas deposits.
In [1] the authors adapt financial portfolio theory as a
method for ecosystem-based fishery management that
accounts for species interdependencies, uncertainty, and
sustainability constraints. They illustrate their method with
routinely collected data available from the Chesapeake Bay
and demonstrate the gains from taking into account variances
and covariances of gross fishing revenues in setting species
total allowable catches.
In [16] the authors embed a portfolio decision framework
into a multi-period bio-economic model in order to quantify
the risk-benefit tradeoffs of alternative strategies. They
develop alternative sets of processed seafood products for
managing the risks that occur as a result of harvests from
commercial fish stocks. The model is used to generate an
efficient portfolio frontier to estimate possible rent dissipation
from status quo management. Frontiers are also generated for
seafood processors and brokers. The authors discuss
implications for the different industry agents.
Drawing the analogy between managing risky assets and
managing multispecies fisheries is a relatively new idea, even
though the foundation for this idea is neither new to ecology
nor economics. In ecology, Walters at al. [17] derives a meanvariance frontier for single-species management, while Real
[18] uses portfolio theory to describe animal behavior.
Portfolio management of fisheries can be a means of
allocating catch across life history stages. Baldursson and
Magnusson [19], Arrason [20] alludes to multispecies
portfolio management in a deterministic bioeconomic model
by suggesting that managers choose a vector of Total
Allowable Catches (TACs), while Hanna [21] explicitly
discusses the idea of selecting “species portfolios” as a means
to match management objectives with ecosystem structure.
Hilborn et al. [22] provide a justification for portfolio
management at the regional level by pointing out that total
productivity aggregated across species is subject to less
variability than the productivity of individual species.
Edwards et al. [15] formally develop the analogy within the
context of standard bio-economic models and provide a
stylized simulation of a three-species system. More recently,
Perusso at al. [23] apply portfolio theory to individual
fishermen targeting decisions in the U.S. Atlantic and Gulf of
Mexico pelagic longline fishery. An interesting survey on the
application of portfolio theory to fisheries management may
be found in [24]. Several references on applications of
operations research to fish management can be found in [25],
[26].
III. MEAN-VARIANCE PORTFOLIO SELECTION MODELS FOR
2
FISHERIES MANAGEMENT
Suppose that we have n fish species S1 , S 2 ,..., S n . Denote
by ξ i the return obtained from one kg of fish of species S i .
Of course all ξ i are random variables. Let ξ = (ξ 1 , ξ 2 ,..., ξ n )
denote the random vector of returns. Denote by μ i the mean
of ξ i . Let μ = (μ1 , μ 2 ,..., μ n ) . Denote by bi the cost
associated to fishing one kg of fish from species S i and with
a i the maximum sustainable yield for the fish of species S i .
a = (a1 , a 2 ,..., a n ) , b = (b1 , b2 ,..., bn ) . Denote by
Let
C = cij the covariance matrix of the random vector
( )
(
)
ξ = (ξ 1 , ξ 2 ,..., ξ n ) . That is cij = cov ξ i , ξ j , 1 ≤ i, j ≤ n . The
signification of c ij
is the correlation between species
revenues. These correlations can be negative or positive
depending on the relative strength of trophic interactions,
environmental fluctuations, and fishing intensity and gear
293
Proceedings of the International Conference on Applied Computer Science
choices that determine fish stocks and corresponding catch
rates, as well as output market interactions that affect prices.
Let x i be the quantity harvested from fish of species S i .
Of course x i ≥ 0 for every i=1,2,…,n. The vector
x = (x1 , x 2 ,..., x n ) will be called a portfolio. The cost of
n
harvesting a portfolio x is equal to
∑b x
i
i
B. The maximum expected return problem
In the frame of this problem the decision maker looks for a
portfolio x = (x1 , x 2 ,..., x n ) that maximizes the expected return
and satisfies the following constraints: the risk is smaller than
a given limit τ and the sum invested in the portfolio of fish is
equal to M.
⎧max μ T x
⎪ T
⎨x Cx ≤ τ
⎪ T
⎪⎩b x = M , 0 ≤ x ≤ a
An important problem is to find the range of the parameter
τ . In order to find the limits of this range we shall solve the
following problems
( τ min) min x T Cx : b T x = M ,0 ≤ x ≤ a
5
= b T x . The return
i =1
(resp. the expected return) from harvesting a portfolio x is
n
equal to
∑ξ x
i
i
= ξ T x (resp. to
i =1
n
∑μ x
i i
= μ T x ).
i =1
In reality the vectors a = (a1 , a 2 ,..., a n ) , ξ = (ξ 1 , ξ 2 ,..., ξ n )
and μ = (μ1 , μ 2 ,..., μ n ) and the matrix C = c ij depend on
( )
{
}
( τ max) max{ x Cx : b x = M ,0 ≤ x ≤ a}
time. Fishing and pollution are important factors that make
that the above mentioned vectors and matrix vary along the
time. For short periods of time one can consider that these
vectors and the covariance matrix are constant.
The financial risk of the portfolio x may be defined in
several ways. One traditional way is to define the risk as the
variance of the return f 1 (x ) = Var ξ T x = x T Cx .
A general problem in the fisheries management is the
following multi-objective problem
⎧minimize x T Cx
⎪
T
⎪maximize μ x
⎨
⎪subject to
⎪b T x = M , 0 ≤ x ≤ a
⎩
Starting from the above multi-objective problem one can
formulate the following single objective problems: the
minimum risk problem, the maximum expected return
problem, the optimal tradeoff problem.
T
Denote the optimal value of the problem ( τ min) with τ 1
and the optimal value of the problem ( τ max) with τ 2 . The
decision maker may choose the parameter τ in the range
[τ 1 ,τ 2 ] .
( )
C. The optimal tradeoff problem
6
For every α ∈ [0,1] consider the problem
( )
( )
{
D. Numerical example
In this section we shall study an application of the minimum
risk model for a fish farm of semi-intensive type, located in
the region Danube Delta, at village Jurilovca, Tulcea county,
Romania. The fishery area is 834 ha and 779 ha of it are
covered by water. We study 9 fish species: Carp, Sanger,
Novac, Pike, Catfish, Crucian, Perch, Bream and Roach. The
historical data on the fish market prices are from the period
2000-2008. The sum invested in the harvest plan is M=5000
RON. As a result of computation we find W min = 2.71 x 103
RON,
W max = 21.81 x 103 RON.
7
4
)
TABLE 1. THE MAXIMUM SUSTAINABLE YIELD (MSY) AND THE
HARVESTING COST FOR THE FISH SPECIES
Fish Type
MSY
Harvesting cost
Carp
One can easily see that 0 ≤ M ≤ b T a . An important problem
is to find the range of the parameter W. In order to find the
limits of this range we shall solve the following problems
(Wmin) min μ T x : b T x = M ,0 ≤ x ≤ a
{
max{μ
}
P(α ) min (1 − α )x T Cx − αμ T x : b T x = M ,0 ≤ x ≤ a
Every optimal portfolio of the above problem is called an
efficient portfolio.
A. The minimum risk problem
In the frame of this problem the decision maker looks for a
portfolio x = (x1 , x 2 ,..., x n ) that minimizes the financial risk
and satisfies the following constraints: the expected return is
greater than a given limit W and the sum invested in the
portfolio of fish is equal to M.
⎧min x T Cx
⎪ T
⎪μ x ≥ W
⎨ T
⎪b x = M
⎪0 ≤ x ≤ a
⎩
(
T
}
x = M ,0 ≤ x ≤ a}
T
x : bT
(Wmax)
Denote the optimal value of the problem (Wmin) with W1
and the optimal value of the problem (Wmax) with W2 . The
decision maker may choose the parameter W in the range
[W1 ,W2 ] .
294
in tons
in RON/tone
1
9120
Sanger
0.5
5920
Novac
0.7
5920
Pike
0.4
3650
Catfish
0.5
3650
Crucian
0.3
3650
Perch
0.3
3650
Bream
0.3
3650
Roach
0.3
3650
Proceedings of the International Conference on Applied Computer Science
TABLE 2. HISTORICAL DATA ON THE MARKET PRICES FOR THE FISH SPECIES (IN 103 RON /TONE)
Fish Type
2000
2001 2002
2003
2004
2005
2006
2007
2008
Carp
4.5
4.5
5.0
5.5
6.2
6.7
7.0
7.5
7.8
Sanger
2.5
2.5
2.8
2.8
3.0
3.2
3.5
3.8
5.0
Novac
2.5
2.5
2.8
2.8
3.0
3.2
3.5
3.5
5.0
Pike
4.5
5.0
5.5
6.5
6.5
7.0
7.0
7.0
7.5
Catfish
6.5
7.0
7.0
8.0
8.5
8.5
9.0
9.0
9.5
Crucian
1.8
2.0
2.2
2.2
2.4
2.6
2.8
3.0
3.5
Perch
9.5
9.5
9.5
10.0
10.0
11.0
11.0
11.0
13.0
Bream
1.8
1.8
2
2.2
2.2
2.5
2.7
2.8
3.2
Roach
1.8
1.8
2
2.2
2.2
2.5
2.7
2.8
3.2
TABLE 3. FISH PORTFOLIOS FOR VARIOUS VALUES OF THE PARAMETER W.
W
Nr.
1
Carp
2.71
0.16
Sanger
Novac
Pike
Catfish
0.00
0.59
0.00
0.00
Crucian
Perch
Bream
0.00
0.00
0.00
Roach
0.00
Fob
0.37
2
3.65
0.16
0.00
0.20
0.00
0.00
0.00
0.06
0.30
0.30
0.42
3
4.60
0.17
0.00
0.11
0.00
0.00
0.00
0.17
0.30
0.30
0.52
4
5.54
0.19
0.00
0.02
0.00
0.00
0.00
0.28
0.30
0.30
0.62
5
6.49
0.13
0.00
0.00
0.00
0.14
0.00
0.30
0.30
0.30
0.75
6
7.44
0.03
0.00
0.05
0.00
0.31
0.00
0.30
0.30
0.30
0.90
7
8.38
0.00
0.00
0.00
0.00
0.47
0.00
0.30
0.30
0.30
1.06
8
9.33
0.00
0.00
0.00
0.20
0.50
0.00
0.30
0.30
0.07
1.30
9
10.27
0.00
0.00
0.00
0.40
0.50
0.00
0.30
0.24
0.00
1.62
10
11.22
0.01
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
2.10
11
12.17
0.17
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
2.66
12
13.11
0.33
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
3.28
13
14.06
0.48
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
3.98
14
15.00
0.64
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
4.74
15
15.95
0.79
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
5.58
16
16.89
0.95
0.00
0.00
0.40
0.50
0.00
0.30
0.30
0.30
6.48
17
17.84
1.00
0.00
0.00
0.40
0.50
0.25
0.30
0.30
0.30
7.45
18
18.79
1.00
0.00
0.26
0.40
0.50
0.30
0.30
0.30
0.30
8.53
19
19.73
1.00
0.00
0.55
0.40
0.50
0.30
0.30
0.30
0.30
9.70
20
20.68
1.00
0.15
0.70
0.40
0.50
0.30
0.30
0.30
0.30
11.00
21
21.62
1.00
0.44
0.70
0.40
0.50
0.30
0.30
0.30
0.30
12.40
14
12
Risk
10
8
6
4
2
0
2.71 3.65 4.6 5.54 6.49 7.44 8.38 9.33 10.3 11.2 12.2 13.1 14.1 15
16
16.9 17.8 18.8 19.7 20.7 21.6
W
Figure 1. The efficient frontier map associated to the minimum risk problem
In the second (resp. third) column from Table 1 is displayed
the vector a (resp. the vector b). In the Table 2 are presented
the historical data of the market prices for the fish species and
in the Table 3, the optimal fish portfolios for different values
295
Proceedings of the International Conference on Applied Computer Science
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Environmental Variability and Uncertain Production Parameters,”
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of parameter W. In the last column of the Table 3 is displayed
the optimal value of the objective function (the variance of the
return).
In Figure 1 is displayed the graph of the efficient frontier
map associated to the minimum risk problem. That is to every
value of the parameter W corresponds the minimum value of
the objective map (the variance of the return).
The minimum risk problem was solved for various values
of the parameter W with the NLP solver from GAMS. The
limits of the range of variation of the parameter W were
obtained with the MINLP solver from GAMS.
3
IV.CONCLUSION
The paper studies an approach based on portfolio theory for
fisheries management. The numerical example presented is
based on real data. It shows that the portfolio theory is a
suitable approach for the fish management. A valuable idea
for further research is to study portfolio selection models
based on downside risk measures such as: semi-variance,
lower partial moments, probability shortfall or value-at-risk.
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